Find Increasing Intervals for the Quadratic Function y = x² + 2x - 8

To find the intervals where the function $y = x^2 + 2x - 8$ is increasing, follow these steps:

• Step 1: Find the derivative of the function. The derivative, $y'$, is obtained by differentiating $y = x^2 + 2x - 8$ with respect to $x$.
  $y' = \frac{d}{dx}(x^2 + 2x - 8) = 2x + 2$.
• Step 2: Find critical points by setting the derivative equal to zero:
  $2x + 2 = 0$
  Solve for $x$ to find $x = -1$.
• Step 3: Determine the sign of $y'$ on the intervals determined by this critical point. We have two intervals to consider: $(-\infty, -1)$ and $(-1, \infty)$.
• For $x < -1$, choose a test point, such as $x = -2$:
  $y'(-2) = 2(-2) + 2 = -4 + 2 = -2$ (negative, so $y$ is decreasing).
• For $x > -1$, choose a test point, such as $x = 0$:
  $y'(0) = 2(0) + 2 = 2$ (positive, so $y$ is increasing).

The function is increasing on the interval $x > -1$.

Therefore, the solution to the problem is $x > -1$.